next up previous contents
Next: Model Selection Up: The Distribution Functions and Previous: Solving for the Distribution

The Equations of Motion for a Binary Fluid

It now must be shown that any binary mixture with these equilibrium distributions satisfies the correct hydrodynamic equations. This is done here using the standard Chapman-Enskog method. A similar expansion approach is employed in reference [38] using a perturbation parameter tex2html_wrap_inline14741 , the time between collisions. Equations (4.105) and (4.106) can be Taylor expanded:

  equation3603

and

  equation3624

Expanding the distribution functions and the time and space derivatives:

  equation3645

and using equations (4.150) and (4.151) we can perform a Chapman-Enskog expansion. Substituting equation (4.152) into equations (4.150) and (4.151) gives

  equation3666

and

  equation3701

It is still require that the total mass and momentum and the mass of each component are conserved at each site. This is achieved, to second-order, if the zeroth-order expansion of the distribution functions are equal to their equilibrium values and if

  equation3742

To first-order in tex2html_wrap_inline13299 equation (4.153) is

  equation3751

Summing equation (4.156) over i gives the first-order continuity equation for the total population:

  equation3763

Multiplying equation (4.156) by tex2html_wrap_inline14187 and summing over i gives the first-order continuity equation for the fluid momentum:

  equation3771

Similarly equation (4.154) to first-order in tex2html_wrap_inline13299 is

  equation3782

Summing over i gives the first-order convection-diffusion equation

  equation3796

To second-order in tex2html_wrap_inline13299 equation (4.153) is

  equation3803

The first and third terms sum to zero by equation (4.155) and the fourth and fifth terms also sum to zero as in equations (4.157) and (4.158) respectively. Thus summing equation (4.161) we obtain the second-order continuity equation for the whole fluid:

equation3838

Combining this with the first-order equation (4.157) and recombining the derivatives we get the continuity equation for the whole fluid

equation3842

Multiplying equation (4.161) by tex2html_wrap_inline14193 and summing over i gives

  equation3848

To find expressions for the two summations in equation (4.164) consider first

equation3873

With tex2html_wrap_inline14777 and the sum of all terms containing an odd number of tex2html_wrap_inline13399 s being zero this gives

  equation3896

Next consider equation (4.156) multiplied by tex2html_wrap_inline14781 and summed over i:

  equation3919

Substituting equations (4.166) and (4.167) into equation (4.164) gives

  equation3945

We are dealing here with terms tex2html_wrap_inline14785 so, since tex2html_wrap_inline12875 is also small, we can neglect terms tex2html_wrap_inline14035 . From the definition of tex2html_wrap_inline13125 , equations (4.114) and (4.115), we also see that, neglecting higher-order derivatives, we can write tex2html_wrap_inline14793 [38]. We can now re-write the term in square brackets in equation (4.168)

equation3975

Using the expression for tex2html_wrap_inline14795 from equation (4.157) we can combine equations (4.158) and (4.168) to obtain the Navier-Stokes equation

equation3992

where

  equation4005

To second-order in tex2html_wrap_inline13299 equation (4.154) is

equation4018

Summing over i and noting that the first and fourth terms are zero we can write

  equation4054

Multiplying equation (4.159) by tex2html_wrap_inline14801 and summing over i we find the term tex2html_wrap_inline14805 :

equation4084

Substituting this into equation (4.173) we get

  equation4102

where as before we have neglected terms smaller than tex2html_wrap_inline14807 . Finally we can express tex2html_wrap_inline14809 as tex2html_wrap_inline14811 [38] and perform the differentiation on both the terms in brackets:

equation4119

Replacing the time derivatives with spatial derivatives using equations (4.157), (4.158) and (4.160) and, as before, neglecting terms tex2html_wrap_inline14035 gives

  equation4137

for the final term in equation (4.175). Combining equation (4.175) with equation (4.160) and neglecting terms of order tex2html_wrap_inline14815 we obtain the convection-diffusion equation

equation4147

where

equation4158

The Navier-Stokes equation for the liquid-gas model follows [63, 38] in the same manner as the Navier-Stokes equation for the binary fluid. The kinematic and bulk viscosities have the same tex2html_wrap_inline13341 dependence as equation (4.171).


next up previous contents
Next: Model Selection Up: The Distribution Functions and Previous: Solving for the Distribution

James Buick
Tue Mar 17 17:29:36 GMT 1998